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%%% Catral, Minerva
\par \vskip 0.2in
\medskip\noindent E-mail:   mcatral@uvic.ca
\section*{The Kemeny Constant in  Finite Homogeneous Ergodic Markov Chains} By {\sl Minerva Catral}.  

\noindent For a finite homogeneous ergodic Markov chain, the Kemeny constant is an interesting quantity which is defined in terms of the mean first passage times and the stationary distribution vector.  A formula in terms of group inverses and inverses of associated M-matrices is presented and  perturbation results are derived.\par
\medskip\noindent Keywords: Kemeny constant, Finite Markov Chains, Group Inverses\par
\medskip\noindent AMS classification: 15%%
%%% Eubanks, Sherod
\par \vskip 0.2in
\medskip\noindent E-mail:   eubanks@math.wsu.edu
\section*{Generalized Soules Matrices} By {\sl Sherod Eubanks}.  

\noindent I will discuss a generalization of Soules matrices and its application to the nonnegative inverse eigenvalue problem, eventually nonnegative matrices, and exponentially nonnegative matrices.\par
\medskip\noindent Keywords: eventually nonnegative matrices, exponentially nonnegative matrices, inverse eigenvalue problem, Soules matrix\par
\medskip\noindent AMS classification: 15A57%15A29%
%%% Laffey, Thomas
\par \vskip 0.2in
\medskip\noindent E-mail:   Thomas.Laffey@ucd.ie
\section*{Some constructive techniques in the nonnegative inverse eigenvalue problem} By {\sl Thomas Laffey}.  

\noindent Let $\sigma:=(\lambda_{1},\quad...\quad,\lambda_{n})$ \ be a list of complex numbers and let  \[ s_{k}:=\lambda_{1}^{k}+\quad...\quad+\lambda_{n}^{k},\quad k=1,2,3,\quad... \]  be the associated Newton power sums. A famous result of Boyle and Handelman states that if all the $s_{k}$ are positive, then there exists a nonnegative integer $N$ such that \[ \sigma_{N}:=(\lambda_{1},\quad...\quad,\lambda_{n},0,\text{ \ \ \ }...\text{ \ \ },0),\text{ \ \ \ \ \ (}N\text{ \ \ zeros)} \] is the spectrum of a nonnegative $(n+N)\times(n+N)$ matrix \ $A$. \ The problem of obtaining a constructive proof of this result with an effective bound on the minimum number $N$ of zeros required has not yet been solved. We present a number of techniques for constructing nonnegative matrices with given nonzero spectrum \ $\sigma$, and use them to obtain new upper bounds on the minimal size of such an $A$, for various classes of $\sigma$.  This is joint work with Helena Smigoc.\par
\medskip\noindent Keywords: Nonegative Matrices, Nonzero Spectrum\par
\medskip\noindent AMS classification: 15%%This is a contrubution to the minisymposium on nonnegative matrices.
%%% McDonald, Judith
\par \vskip 0.2in
\medskip\noindent E-mail:   jmcdonald@math.wsu.edu
\section*{Nonnegative and Eventually Nonnegative Matrices} By {\sl Judith McDonald}.  

\noindent I will discuss the interplay between the properties of nonnegative and eventually nonnegative matrices, and the role that the inverse eigenvalue problem plays in this relationship.\par
\medskip\noindent Keywords: nonnegative, eventually nonnegative, inverse eigenvalue problem\par
\medskip\noindent AMS classification: 15A48%15A18%This is part of the mini-symposium on nonnegative and eventually nonnegative matrices
%%% Neumann, Michael
\par \vskip 0.2in
\medskip\noindent E-mail:   neumann@math.uconn.edu
\section*{On Optimal Condition Numbers For Markov Chains} By {\sl Michael Neumann and Nung--Sing Sze}.  

\noindent Let $T=(t_{i,j})$ and $\tilde{T}=T-E$ be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on $n$ states. A function $\kappa(\cdot)$ is called a {\it condition number for Markov chains}  with respect to the $(\alpha,\beta)$--norm pair if $\|\pi-\tilde{\pi}\|_\alpha \leq \kappa(T)\|E\|_\beta$.\\  Various condition numbers, particularly with respect to the $(1,\infty)$ and $(\infty,\infty)$ have been suggested in the literature by several authors. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient $\tau_p(\ast)=\sup_{y^te=0} \frac{\|y^t\ast\|_p}{\|y\|_1}$, where $e$ is the $n$--vector of all $1$'s, is the smallest of the condition numbers of Markov chains with respect to the $(p,\infty)$--norm pair. We use this result to identify the smallest condition number of Markov chains among the $(\infty,\infty)$ and $(1,\infty)$--norm pairs. These are, respectively, $\kappa_3$ and $\kappa_6$ in the Cho--Meyer list of $8$ condition numbers.\\  Kirkland has studied  $\kappa_3(T)$. He has shown that $\kappa_3(T)\geq\frac{n-1}{2n}$ and he has characterized the properties of transition matrices for which equality holds. We prove again that $2\kappa_3(T)\leq \kappa(6)$ which appears in the Cho--Meyer paper and we characterize the transition matrices $T$ for which $\kappa_6(T)=\frac{n-1}{n}$. There is only one such matrix: $T=(J_n-I)/(n-1)$. where $J_n$ is the $n\times n$ matrix of all $1$'s. This result demands the development of the cyclic structure of a doubly stochastic matrix with a zero diagonal. 
\\  Research supported by NSA Grant  No.  06G--232\par
\medskip\noindent Keywords: Markov chains, stationary distribution, stochastic matrix, group inverses, sensitivity analysis, perturbation theory, condition numbers.\par
\medskip\noindent AMS classification: 15A51%65F35%This talk is for the Nonnegative and Eventually Nonnegative Matrix Mini-symposium.
%%% Morris, DeAnne
\par \vskip 0.2in
\medskip\noindent E-mail:   dmorris@math.wsu.edu
\section*{Jordan forms corresponding to nonnegative and eventually nonnegative matrices} By {\sl Judith McDonald, DeAnne Morris}.  

\noindent We give necessary and sufficient conditions for a set of Jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix. For each eigenvalue, $\lambda,$ the $\lambda$-level characteristic (with respect to the spectral radius) is defined. The necessary and sufficient conditions include a requirement that the $\lambda$-level characteristic is majorized by the $\lambda$-height characteristic. An algorithm which determines whether or not a multiset of Jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix will be discussed. We also offer necessary and sufficient conditions for a multiset of Jordan blocks to correspond to the spectrum of an eventually nonnegative matrix.\par
\medskip\noindent Keywords: nonnegative\par
\medskip\noindent AMS classification: %%
%%% Smigoc, Helena
\par \vskip 0.2in
\medskip\noindent E-mail:   Helena.Smigoc@ucd.ie
\section*{An example of constructing a nonnegative matrix with given spectrum} By {\sl Thomas J. Laffey, Helena {\v S}migoc}.  

\noindent We say that a list of $n$ complex numbers $\sigma$ is the nonzero spectrum of a nonnegative matrix, if there exists a nonnegative integer $N$ such that $\sigma$ together with $N$ zeros added to it is the spectrum of some $(n+N)\times (n+N)$ nonnegative matrix. Boyle and Handelman characterized all lists of $n$ complex numbers that can be the nonzero spectrum of a nonnegative matrix.  In this talk we will present a constructive proof that $\tau(t)=(3+t,3-t,-2,-2,-2)$ is the nonzero spectrum of some nonnegative matrix for every $t >0.$  We will give a bound for the number of zeros that needs to be added to $\tau(t)$ to achieve a nonnegative realization. We will discuss how the method presented could be applied to more general situations.\par
\medskip\noindent Keywords: Nonnegative Inverse Eigenvalue Problem, Nonzero Spectrum, Spectral Gap\par
\medskip\noindent AMS classification: 15A48%15A18%

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